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302 X EXISTENCE THEOREMS

There is then at most one solution of x' =/(/, x) taking a given value x₀ E H for / — t_Qand defined in a neighborhood of t_Q (Problem 5(a)). But in addition, if u, v are two
approximate solutions of x' =/(/, x) in an open ball J of center t_Q , contained in I, with
approximations Si,s₂₉ and such that u(t₀) = v(t₀) = x_0> then, for any / e J

(Use the same method as in (10.5.1).)

(b) Let I = ]— 1 , 1 [, H = E = R, and let ® be the set of all real valued functions /,
continuous in I x H, and such that, for /^ 0 in I, \f(t, x^ — /(/, x₂)\ ^ \xj. — x₂\l\t\.
There is then at most one solution of x^/ — f(t, x) taking a given value for t — t_Q and
defined in a neighborhood of t₀ (Problem 5(a)). But prove that there is no function
cp(t, e) ^ 0 such that, for any pair (#, v) of approximate solutions, with approxima-
tion s, of any equation x^/ = /(/, x) with / e 0, such that u and v are defined in I and
_u(t_Q) = v(t₀), the relation \\u(t) — v(t)\\ ^ q>(\t\, s) would hold for every /el. (For any
a e ]0, 1[, let /be the continuous function equal to x/t for |*| ^ t²/(oc — t)₉ 0 ^ / < a,
and for / ^ a, and independent of x for the other values of (/, x) such that / ^ 0; define
f(t, x) = f(—t_t x) for t =s* 0. Take u = 0; let v(t) = et for \t ^ a, and take for v a
solution of x' =/(*, x) -f e for the other values of /.)

7. The notations being those of Section 10.4, suppose E is finite dimensional and / is
continuous in I x H; let (/₀ , *o) be a point of I x H, J an open ball of center t₀contained in I, S an open ball of center x₀ such that S cz H. Suppose / is bounded in
J x S, and the following conditions are verified:

(1) There is at most one solution of x' =/(/, x) defined in an open interval con-
tained in J and containing t₀ , and taking the value x₀ for t = t_Q .

(2) There exists a sequence (//„)« 3=0 of continuous mappings of J into S such that

u_n(t) = x₀ + /(y, w_n_iW) ds for n ^ 1 and / e J.

^J'o

(3) For every / e J, u_n+i(t) — u_n(t) converges to 0 when n tends to + oo.

Show that in every compact interval J' c J containing *₀ the sequence (#„) converges
uniformly to a solution of xf ~f(t, x) equal to x₀ for t = /₀ . (Observe that the sequence
(«„) is equicontinuous; use Ascoli's theorem (7.5.7), as well as (3.16.4) and (8.7.8).)

8. Suppose E is finite dimensional, oj and / verify the conditions of Problem 5(a), and in
addition, for every t e ]0, a[, the function z -> co(/, z) is increasing in [0, -h oo [. There
is then at most one solution of x' = /(/, x) defined in an interval [0, a[ c: [0, a[ and taking
the value X_Q for / = 0 (Problem 5(a)). Suppose in addition that there exists, in an
interval J = [0, a] ^c [0, a[, a sequence (u_n)_n^₀ of continuous mappings of J into S such
that

u_n(t) = X_Q -H f(s, u_n-i(s)) ds for n ^ 1 and t e J.
Jo

(a) For every teJ, let y_n(t) = \\u_n+1(t)- u_n(t) \\,z_n(t) - sup^_n+fc(0, and w(/) =

fc>0

inf z_n(t). Show that the functions z_n and w are continuous in J (use Problem 11 of

fc>0

Section 7.5).

(b) Let t,t — h be two points of J (h > 0); show that, for every S > 0, there is an N
such that, for n 2* N,

- *,(' ™ h)\ ^ f w(j,
^t-ft



	302 X EXISTENCE THEOREMS There is then at most one solution of x' =/(/, x) taking a given value x₀ E H for / — t_Qand defined in a neighborhood of t_Q (Problem 5(a)). But in addition, if u, v are two approximate solutions of x' =/(/, x) in an open ball J of center t_Q , contained in I, with approximations Si,s₂₉ and such that u(t₀) = v(t₀) = x_0> then, for any / e J

	(Use the same method as in (10.5.1).) (b) Let I = ]— 1 , 1 [, H = E = R, and let ® be the set of all real valued functions /, continuous in I x H, and such that, for /^ 0 in I, \f(t, x^ — /(/, x₂)\ ^ \xj. — x₂\l\t\. There is then at most one solution of x^/ — f(t, x) taking a given value for t — t_Q and defined in a neighborhood of t₀ (Problem 5(a)). But prove that there is no function cp(t, e) ^ 0 such that, for any pair (#, v) of approximate solutions, with approxima- tion s, of any equation x^/ = /(/, x) with / e 0, such that u and v are defined in I and _u(t_Q) = v(t₀), the relation \\u(t) — v(t)\\ ^ q>(\t\, s) would hold for every /el. (For any a e ]0, 1[, let /be the continuous function equal to x/t for \|\| ^ t²/(oc* — t)₉ 0 ^ / < a, and for / ^ a, and independent of x for the other values of (/, x) such that / ^ 0; define f(t, x) = f(—t_t x) for t =s* 0. Take u = 0; let v(t) = et for \t ^ a, and take for v a solution of x' =/(, x) -f e for the other values of /.) 7. The notations being those of Section 10.4, suppose E is finite dimensional* and / is continuous in I x H; let (/₀ , o) be a point of I x H, J an open ball of center t₀contained in I, S an open ball of center x₀ such that S cz H. Suppose / is bounded in J x S, and the following conditions are verified: (1) There is at most one* solution of x' =/(/, x) defined in an open interval con- tained in J and containing t₀ , and taking the value x₀ for t = t_Q . (2) There exists a sequence (//„)« 3=0 of continuous mappings of J into S such that u_n(t) = x₀ + /(y, w_n_iW) ds for n ^ 1 and / e J. ^J'o (3) For every / e J, u_n+i(t) — u_n(t) converges to 0 when n tends to + oo. Show that in every compact interval J' c J containing ₀ the sequence (#„) converges uniformly to a solution of xf ~f(t, x)* equal to x₀ for t = /₀ . (Observe that the sequence («„) is equicontinuous; use Ascoli's theorem (7.5.7), as well as (3.16.4) and (8.7.8).) 8. Suppose E is finite dimensional, oj and / verify the conditions of Problem 5(a), and in addition, for every t e ]0, a[, the function z -> co(/, z) is increasing in [0, -h oo [. There is then at most one solution of x' = /(/, x) defined in an interval [0, a[ c: [0, a[ and taking the value X_Q for / = 0 (Problem 5(a)). Suppose in addition that there exists, in an interval J = [0, a] ^c [0, a[, a sequence (u_n)_n^₀ of continuous mappings of J into S such that u_n(t) = X_Q -H f(s, u_n-i(s)) ds for n ^ 1 and t e J. Jo (a) For every teJ, let y_n(t) = \\u_n+1(t)- u_n(t) \\,z_n(t) - sup^_n+fc(0, and w(/) = fc>0 inf z_n(t). Show that the functions z_n and w are continuous in J (use Problem 11 of fc>0 Section 7.5). (b) Let t,t — h be two points of J (h > 0); show that, for every S > 0, there is an N such that, for n 2* N,

	- ,(' ™ h)\ ^* f w(j, ^t-ft